Interaction with an obstacle in the 2d focusing nonlinear Schrödinger equation
IINTERACTION WITH AN OBSTACLE IN THE 2D FOCUSINGNONLINEAR SCHR ¨ODINGER EQUATION
OUSSAMA LANDOULSI, SVETLANA ROUDENKO, AND KAI YANG
Abstract.
We present a numerical study of solutions to the 2 d focusing nonlinear Schr¨odinger equationin the exterior of a smooth, compact, strictly convex obstacle, with Dirichlet boundary conditions withcubic and quintic powers of nonlinearity. We study the effect of the obstacle on solutions traveling towardthe obstacle at different angles and with different velocities. We introduce a concept of weak and stronginteractions and show how the obstacle changes the overall behavior of solutions. Contents
1. Introduction 12. The numerical method 72.1. The numerical scheme 72.2. Mass and energy conservation 93. The NLS R equation 124. Perturbations of the soliton 135. Dependence on the distance 156. Weak interaction with an obstacle 176.1. The L -critical case 176.2. The L -supercritical case 227. Strong interaction with an obstacle 257.1. The L -critical case 257.2. The L -supercritical case 278. Conclusion 28References 291. Introduction
We consider the 2 d focusing nonlinear Schr¨odinger equation (NLS Ω ) outside of a strictly convex obstaclewith Dirichlet boundary conditions: i∂ t u + ∆ Ω u = −| u | p − u ( t, x ) ∈ R × Ω ,u ( t , x ) = u ( x ) ∀ x ∈ Ω ,u ( t, x ) = 0 ( t, x ) ∈ R × ∂ Ω , (NLS Ω ) Mathematics Subject Classification.
Key words and phrases.
Focusing NLS equation, convex obstacle, exterior domain, soliton-obstacle interaction, scattering,blow-up. a r X i v : . [ m a t h . A P ] F e b O. LANDOULSI, S.ROUDENKO, AND K.YANG where t ∈ R is the initial time, Ω is an exterior domain in R and ∆ Ω is the Dirichlet Laplace operatordefined by ∆ Ω := ∂ x + ∂ y , ( x, y ) ∈ R . Here, u is a complex-valued function, u : R × Ω −→ C ( t, x ) (cid:55)−→ u ( t, x ) . We consider u ∈ H (Ω) , where the Sobolev space H (Ω) is the set of functions in H (Ω) that satisfyDirichlet boundary conditions, i.e., u = 0 on ∂ Ω.The NLS Ω equation is locally well-posed in H (Ω) in dimension d = 2 , see [9], [8] for a non-trappingobstacle and [22], [21] for a strictly convex obstacle. The solution u can be extended to a maximal timeinterval I = ( − T − , T + ) of existence and the following alternative holds:either T + = ∞ (respectively, T − = ∞ ), or T + < ∞ (respectively, T − < ∞ ) withlim t → T + (cid:107) u ( t, · ) (cid:107) H (Ω) = ∞ (cid:18) respectively, lim t → T − (cid:107) u ( t, · ) (cid:107) H (Ω) = ∞ (cid:19) . During their lifespans, solutions to the nonlinear Schr¨odinger equation outside an obstacle conserveboth mass and energy: M Ω [ u ( t )] := (cid:90) Ω | u ( t, x ) | dx = M Ω [ u ] , (1.1) E Ω [ u ( t )] := (cid:90) Ω |∇ u ( t, x ) | dx − p + 1 (cid:90) Ω | u ( t, x ) | p +1 dx = E Ω [ u ] . (1.2)Unlike the nonlinear Schr¨odinger equation NLS R d posed on the whole Euclidean space R d , the NLS Ω equation does not preserve the momentum P Ω [ u ] = Im (cid:82) Ω ¯ u ( t, x ) ∇ u ( t, x ) dx, since the derivative of themomentum P Ω with respect to the time variable is equal to a non-zero boundary term.Furthermore, the NLS R equation, posed on the whole Euclidean space R , is invariant under thescaling transformation, that is, if u ( t, x ) is a solution to the NLS R equation, then λ p − u ( λx, λ t ) is alsoa solution for λ > . This scaling identifies the critical Sobolev space ˙ H s c x , where the critical regularity s c is given by s c := p − p − . The equation, when s c = 0 , is referred to as the mass-critical (or the L -critical), and when 0 < s c < , iscalled the mass-supercritical (or L -supercritical) and energy-subcritical (or H -subcritical). Throughoutthis paper, we will consider the 2 d cubic ( p = 3) and quintic ( p = 5) NLS Ω equations. Since the presenceof the obstacle does not change the intrinsic dimensionality of the problem, we may regard the cubicNLS Ω equation as being the mass-critical equation and the quintic one as the mass-supercritical andenergy-subcritical (or intercritical) equation.The focusing NLS equation, posed on the whole space, admits soliton solutions that are periodic intime, that is, u ( t, x ) = e itω Q ω ( x ) , where ω > Q ω is an H smooth solution of the nonlinear ellipticequation, − ∆ Q ω + ω Q ω = | Q ω | p − Q ω . (1.3)In this paper, we denote by Q ω the ground state solution, that is, the unique, positive, vanishing atinfinity H solution of (1.3). The ground state solution turns out to be radial, smooth and exponentiallydecaying function (for s c < Q ω characterized as the unique min-imizer for the Gagliardo-Nirenberg inequality up to scaling, space translation and phase shift, see [28]. D NLS OUTSIDE AN OBSTACLE 3
For simplicity, we denote by Q the ground state solution of (1.3), when ω = 1 . The NLS equation, posed on the whole Euclidean space R d , also enjoys Galilean invariance: if u ( t, x )is a solution, then so is u ( t, x − vt ) e i ( x · v − | v | t ) , v ∈ R d . Applying the Galilean transform to the solution e itω Q ω ( x ) of the NLS on R d , we obtain a soliton solution, moving on the line x = tv with a velocity v ∈ R d : u ( t, x ) = e i ( ( x · v ) − | v | t + t ω ) Q ω ( x − t v ) . (1.4)The soliton solution is a global solution of the focusing NLS equation, but it is not a soliton solutionfor the NLS Ω equation: this soliton solution does not satisfy the Dirichlet boundary conditions.In [31], the first author constructed a solitary wave solution for the 3 d focusing L -supercritical NLS Ω equation for large t, which behaves asymptotically as a soliton on the Euclidean space R , traveling witha velocity v, and moving away from the obstacle. Indeed, let T > , c ω > C ∞ functionsuch that Ψ = 0 near the obstacle and Ψ = 1 for | x | (cid:29)
1, then (cid:13)(cid:13)(cid:13) u ( t, x ) − e i ( ( x · v ) − | v | t + t ω ) Q ω ( x − tv )Ψ( x ) (cid:13)(cid:13)(cid:13) H (Ω) ≤ e − c ω | v | t ∀ ( t, x ) ∈ [ T , + ∞ ) × Ω , is a solution of the NLS Ω equation.This solution is global, it does not scatter. This proves the optimality of the following threshold forthe global existence and scattering given in [27] for the cubic NLS Ω equation, in dimension d = 3: let u ∈ H (Ω) satisfy E Ω [ u ] M Ω [ u ] < E R [ Q ] M R [ Q ] , (1.5) (cid:107) u (cid:107) L (Ω) (cid:107)∇ u (cid:107) L (Ω) < (cid:107) Q (cid:107) L ( R ) (cid:107)∇ Q (cid:107) L ( R ) . (1.6)Then the solution u ( t ) scatters in H (Ω) in both time directions.This threshold was first proved for the 3 d cubic NLS equation on the whole space R by the secondauthor in [20] with Holmer (in the radial setting) and in [14] with Duyckaerts and Holmer (nonradialcase); further generalizations can be found in [17], [19].In [15], the first two authors with Duyckaerts studied the dynamics of the focusing 3 d cubic NLS Ω equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when E Ω [ u ] M Ω [ u ] = E R [ Q ] M R [ Q ] , with the initial mass-gradient bound on u ∈ H (Ω), (cid:107) u (cid:107) L (Ω) (cid:107)∇ u (cid:107) L (Ω) < (cid:107) Q (cid:107) L ( R ) (cid:107)∇ Q (cid:107) L ( R ) , where Q is the ground state solution of (1.3), with ω = 1. The same problem was studied in the whole Eu-clidean space by Duyckaerts and the second author in [16] for the focusing cubic NLS equation on R . Thedynamics of the NLS equation on the whole Euclidean space is more involved. Indeed, the authors provedthat if the initial datum u ∈ H ( R ) satisfies the same mass-gradient condition as above, then the solu-tion u ( t ) scatters or u ( t ) is a “special solution” Q + , up to symmetries, that scatters in negative time andconverges to the soliton e it Q (up to symmetries) in positive time. We show in [15] that this special solutiondoes not have an analogue for the problem in the exterior of an obstacle and prove that such solutionsare globally defined and scatter in the positive time direction. The existence of blow-up solutions at themass-energy threshold for the NLS equation on the whole space was also proved in [16] and the behavior of O. LANDOULSI, S.ROUDENKO, AND K.YANG solutions is related to another special solution Q − . It was proved that if E R [ u ] M R [ u ] = E R [ Q ] M R [ Q ]and (cid:107) u (cid:107) L ( R ) (cid:107)∇ u (cid:107) L ( R ) > (cid:107) Q (cid:107) L ( R ) (cid:107)∇ Q (cid:107) L ( R ) , then the solution u ( t ) blows up in finite time or u ( t ) is a special solution Q − , up to symmetries. The existence of blow-up solutions at the mass-energythreshold E Ω [ u ] M Ω [ u ] = E R [ Q ] M R [ Q ] and (cid:107) u (cid:107) L (Ω) (cid:107)∇ u (cid:107) L (Ω) > (cid:107) Q (cid:107) L ( R ) (cid:107)∇ Q (cid:107) L ( R ) , for theNLS Ω equation is currently an open question.All results obtained for the NLS Ω equation are for the globally existing and scattering solutions, how-ever, the existence of blow-up solutions has been an open question for several years. The classical proofby the convexity argument on the Euclidean space R d fails in the exterior of an obstacle due to the ap-pearance of the boundary terms with an unfavorable sign in the second derivative of the variance V( u ( t )),that is, if V( u ( t )) := (cid:90) R d | x | | u ( t, x ) | dx, (1.7)then 116 d dt V( u ( t )) = E [ u ] − (cid:18) d − d + 2 p + 1 (cid:19) (cid:90) Ω | u | p +1 dx − (cid:90) ∂ Ω |∇ u | ( x · (cid:126)n ) dσ ( x ) , (1.8)where (cid:126)n is the unit outward normal vector. One can see that in the last term x · (cid:126)n ≤ , for all x ∈ ∂ Ω . Recently, the first author in [30] (see also [29]) has proved the existence of blow-up solution to theNLS Ω equation in the exterior of a smooth, compact, convex obstacle. This was the first step in the studyof the existence of blow-up solutions to (NLS Ω ). We introduced a new modified variance V ( u ( t )) , whichis bounded from below and is strictly concave for the solutions that we consider, namely, V ( u ( t )) := (cid:90) Ω ( d ( x, Ω c )) | u ( t, x ) | dx, (1.9)where d ( x, Ω c ) = | x | − R is the distance to the obstacle and R is the radius of the obstacle (disk).In [30] (see also [29]), it was proved that solutions with finite variance and negative energy blow up infinite time. Furthermore, it was shown that finite variance solutions to the NLS Ω equation for p ≥ d , which satisfy (1.5), (cid:107) u (cid:107) L (Ω) (cid:107)∇ u (cid:107) L (Ω) > (cid:107) Q (cid:107) L ( R ) (cid:107)∇ Q (cid:107) L ( R ) and a certain symmetry condition, willblow up in finite time.From the above review one notices that an obstacle does influence the behavior of solutions and whilesome properties and criteria remain robust and almost unchanged (except for the restriction of the wholespace to the exterior domain problem Ω), other properties either get significantly modified or even more,become unclear in the obstacle setting.Further theoretical study of the interaction between solutions and an obstacle is needed, though thepresence of the obstacle breaks down quite a few properties of the equation, which poses additional diffi-culties for analytical investigations. The purpose of this paper is to investigate this question numericallyto gain further insight of the obstacle influence.In our simulations, we distinguish two types of interaction between a solitary wave solution moving onthe line (cid:126)x = t(cid:126)v ( v is the velocity vector) and the obstacle: Strong interaction:
We call the interaction strong , when a soliton-type solution is moving, towardsthe obstacle, in the same direction as the outward normal (cid:126)n vector of the obstacle, i.e., the velocity vectoris collinear to the normal vector, (cid:126)v = α(cid:126)n, α >
0, (e.g., see Figure 27). In this case, after the collision or
D NLS OUTSIDE AN OBSTACLE 5 the shock, the solitary wave solution does not preserve the shape of the initial or the original soliton butthe solution splits into several solitons or bumps, with a substantial amount of backward reflected waves.
Weak integraction:
We call the interaction weak , when the velocity vector of the moving solitonsolution, towards the obstacle, is not in the same direction as the outward normal vector, i.e., the solutionhits the obstacle at an angle 0 < θ ≤ π between the velocity vector and the outward normal vector, seeFigure 15. In this case, after the interaction, the solitary wave solution is transmitted almost with thesame shape and with insignificant size backward reflected waves.The interaction between a solitary wave-type solution and the obstacle does not depends only on thedirection of the velocity vector and the angle of the collision, it also depends on the initial distancebetween the solitary wave solution and the obstacle. For that, we also investigate the dependence on thedistance. Throughout this paper, we denote by d ∗ := min x ∈ supp (u ) dist( x, Ω c ) , (1.10)the distance between the obstacle Ω c and the essential support of the initial data u such that u iswell-defined, i.e., u satisfies Dirichlet boundary conditions. Note that, if we consider the initial conditionwith the distance d >> d ∗ , then the presence of the obstacle does not affect the behavior of the solutionmuch (provided that u has essentially compact support). For example, if we consider u with a largemass such that d >> d ∗ , then the solution will blow-up in finite time before it could reach the obstaclefor all velocity directions, see Figures 8 and 10 for different scenarios. Therefore, there is no interactionbetween the obstacle and the solution. Moreover, the computation of the value of the boundary termson (1.8) vanishes to 0 , when d >> d ∗ and one can see that the expression of the second derivative of thevariance (1.8) is closer to the one on whole space R . In this case, numerically that soliton behaves as asolution posed on a computational domain without an obstacle, see Figures 4 and 3.For the purpose of this work and in order to study the influence and the interaction of a generic solution(a solitary wave-type solution) with the obstacle, we always consider the distance d to be the minimaldistance d ∗ such that even slight modifications of the velocity direction or the translation parameters,would produce at least a weak interaction.In this paper, we present our numerical funding of the behavior of solutions influenced by an obstaclein the NLS Ω equation outside of a ball, in dimension d = 2 . Our goal is to understand the interactionbetween a solitary wave (for example, traveling with a velocity v ) and the obstacle, as well as the influenceof the obstacle on the nonlinear dynamics of the NLS Ω equation. Here, we study the existence of blow-upsolutions to the NLS Ω equation, in dimension d = 2 . We also investigate the influence of the obstacleon the behavior of blow-up solutions, with different types of interaction that depend on the direction ofthe velocity v and the angle at the collision. According to our numerical simulations, the solitary waveamplitudes decrease at the collision or at any interaction (even a small interaction) between the solitonand the obstacle. This could be explained by the appearance of reflection (or reflection waves) due tothe Dirichlet boundary conditions at the obstacle. After the collision, our numerical results show that, ifthere is a weak or small interaction, then the solitary wave is transmitted almost completely with littlebackward reflection. If there is a strong interaction, then the solution does not preserve the shape ofthe original solitary wave, and it will split the original wave into several waves, continuing as a sum oftwo or more solitary waves with backward reflection. We also observe that the leading reflected wavehas a dispersive behavior. The reflection phenomenon, the loss of the amplitude and the shape of thesolitary wave make the existence of blow-up solutions much more challenging. Nevertheless, we confirm O. LANDOULSI, S.ROUDENKO, AND K.YANG numerically the existence of blow-up solutions after the collision for the 2 d focusing NLS Ω equation inseveral cases of the weak interaction with the obstacle.In addition, we study the sharp threshold for global existence and blow-up solutions to the focusingmass-critical NLS Ω equation in dimension d = 2 . This threshold was first studied by Weinstein in [42] forthe focusing mass-critical NLS equation in the whole Euclidean space R d , for example, p = 3 in dimension d = 2 . He showed a sharp threshold for global existence using Gagliardo-Nirenberg inequality combinedwith the energy conservation, (cid:107)∇ u (cid:107) L ≤ (cid:32) − (cid:107) u (cid:107) L (cid:107) Q (cid:107) L (cid:33) − E [ u ] , which implies that (i) if (cid:107) u (cid:107) L < (cid:107) Q (cid:107) L , then an H solution exists globally in time and (ii) if (cid:107) u (cid:107) L ≥ (cid:107) Q (cid:107) L , then the solution may blow up in finite time. Recently, Dodson proved in [13] thatinitial data u ∈ L ( R d ) with (cid:107) u (cid:107) L < (cid:107) Q (cid:107) L , generates a corresponding solution that is global andscatters in L ( R d ) . In physics, the study of the reflected, diffracted or scattered (mechanical, electromagnetic or gravita-tional) waves after encountering an object, an obstacle or a body, is related to the study of boundaryvalue problems. These problems are usually described mathematically as an exterior domain or obstacleproblem for the wave-type equations with Dirichlet or Neuman boundary conditions. The study of thewave-type equations in the exterior of an obstacle started in the late 1950s and early 1960s and until nowthe understanding of the dynamics of the evolution equations on exterior domains is a widely open areafor investigations. Let us mention, some relevant works on the wave-type equation in an exterior domain.H. W. Calvin and Morawetz have studied the local-energy decay of the solutions to the linear wave equa-tion in an exterior of a sphere and star-shape obstacles, with Dirichlet and Neuman boundary conditions,see [43] and [34], [35]. For later works see [32], [33]. Different results were obtained for almost-star shape,non-trapping and moving obstacles, see [24], [36], [37] and [11]. In that period of time, the authors considera classical solution with C initial data. In 2004 , the Cauchy theory in H (Ω) for the NLS Ω equation wasinitiated by Burq, G´erard and Tzvetkov in [9], for a non-trapping obstacle. After that, the well-posednessproblem for the NLS Ω equation was investigated by others, see for example, [3], [23], [38], [22], [8]. In [31],the first author proved the local well-posedness for the 3 d NLS Ω equation in the critical Sobolev spaceusing the fractional chain rule in the exterior of a compact convex obstacle given in [26].This paper is organized as follows: in Section 2 we present the numerical method that we design forthis study. In Section 3, we show several numerical simulations of scattering and blow-up solutions for thefocusing nonlinear Schr¨odinger equation on the whole Euclidean space R for later comparison. In Section4, we study the sharp threshold for global existence and blow-up solutions for the critical NLS Ω equationin terms of the ground state perturbations. In Section 5, we study the dependence of the interaction onthe initial distance between the solution and the obstacle. In Sections 6 and 7, we study the weak andstrong interactions between the traveling solutions and the obstacle. In all our simulations we considerboth the cubic (mass-critical) and quintic (mass-supercritical) NLS Ω equation. Acknowledgments.
All authors would like to thank Thomas Duyckaerts for fruitful discussions onthis problem. Most of the research on this project was done while O.L. was visiting the Department ofMathematics and Statistics at Florida International University, Miami, FL, during his PhD training. Hethanks the department for hospitality and support. The initial numerical investigations started when K.Y.visited T. Duyckaerts at IHP and LAGA, Paris-13. S.R. and K.Y. were partially supported by the NSFgrant DMS-1927258, and part of O.L.s research visit to FIU was funded by the same grant DMS-1927258(PI: Roudenko).
D NLS OUTSIDE AN OBSTACLE 7 The numerical method
The numerical scheme.
Various numerical methods are employed in order to approximate thenonlinear Schr¨odinger equation ranging from the explicit and implicit schemes in time to the finite Fouriertransform or pseudo-spectral methods in space. There are different methods for the time discretization,for example, the Crank-Nicholson scheme [12], Runge-Kutte type [1], [2], [25], symplectic and splittingtype, [40], [39] and [41], [7] and relaxation methods [5] and [6].In this paper, we use the well-known Crank-Nicholson scheme for the time discretization for the NLS Ω equation. The scheme is based on a time centering approximation u n + ≈ u n +1 + u n . The Crank-Nicholson-type scheme is the 2nd order implicit method. On the plus side, this scheme preserves both the discretemass and the discreet energy exactly during the time evolution. On the negative side, the schemes haveto deal with solving the resulting nonlinear algebraic system, and consequently, the iteration is requiredin solving the nonlinear system at each time step.In this paper we consider only exponentially decaying data, nevertheless, we take a large enoughcomputational domain to approximate the convex domain Ω containing an obstacle. To be specific, weconsider the polar coordinates ( r, θ ) with 0 < r (cid:63) < r < R and 0 ≤ θ ≤ π and use the following domainin our simulations: Ω = { ( r, θ ) ∈ R : r (cid:63) ≤ r ≤ R, ≤ θ ≤ π } . The obstacle is the while region or the disk [0 , r (cid:63) ] × [0 , π ], see Figure 1. It is easy to see that to approx-imate our model, we need to impose the Dirichlet boundary condition on the variable r , i.e., u ( r (cid:63) , θ, t ) = u ( R, θ, t ) = 0, and the periodic boundary conditions on the variable θ , i.e., u ( r, , t ) = u ( r, π, t ). Figure 1.
The computational domain Ω . We next describe our algorithm in detail. We first consider the time discretization. Let T max be theexistence time of the solution and T ∆ t be the computational time ( T ∆ t < T max ). We use N points forthe time discretization, thus, defining a time step ∆ t = T ∆ t N . We discretize the NLS Ω equation at times t n = n ∆ t, n = 0 , .., N by considering the semi-discretization in time u n ≈ u ( x, t n ) with u := u . Thisyields the following time evolution: i u n +1 − u n ∆ t + 12 ∆ u n +1 + 12 ∆ u n = − F ( u n +1 , u n ) , (2.1) O. LANDOULSI, S.ROUDENKO, AND K.YANG where F is the nonlinear term | u | p − u approximated by F ( u n +1 , u n ) := 2 p + 1 (cid:12)(cid:12) u n +1 (cid:12)(cid:12) p +1 − | u n | p +1 | u n +1 | − | u n | u n +1 + u n . (2.2)Note that, u n is the known variable and for n = 0 , u = u is the given initial condition. We computethe evolution u n −→ u n +1 by solving the above system (2.1). For that, we use the Newton iterationto solve the nonlinear implicit system (2.1). We denote u n +1 at the iteration l by u n +1 ,l and assume u n +1 = u n +1 , ∞ , which gives (cid:40) u n +1 ,l +1 = u n +1 ,l − J − · G ( u n +1 ,l ) ,u n +1 , = 1 . · u n , (2.3)where G ( u n +1 ) = u n +1 − u n + ∆ t i ∆ u n +1 + ∆ t i ∆ u n + F ( u n +1 , u n ) and J is the Jacobian of G .The stopping criterion for (2.3) is (cid:13)(cid:13) u n +1 ,l +1 − u n +1 ,l (cid:13)(cid:13) L ∞ < T ol for some small constant T ol . In oursimulation, we take
T ol < − , which is close to the machine precision. In order, to reach the blow-uptime (or the closest time), we slightly decrease the tolerance according to the examples treated.We use the polar transformation in space x = r cos θ and y = r sin θ to convert the problem into thepolar coordinate setting. Thus, we write the Laplacian in polar coordinates as∆ u ( r, θ ) = ∂∂r (cid:18) r ∂∂r u ( r, θ ) (cid:19) + 1 r ∂ ∂θ u ( r, θ ) , ( r, θ ) ∈ Ω . (2.4)We then rewrite the NLS Ω equations for t ∈ (0 , T ) , ( r, θ ) ∈ Ω , as i ∂∂t u ( t, r, θ ) + ∂∂r (cid:18) r ∂∂r u ( t, r, θ ) (cid:19) + 1 r ∂ ∂θ u ( t, r, θ ) = −| u ( t, r, θ ) | p − u ( t, r, θ )with the periodic boundary condition on θu ( t, r,
0) = u ( t, r, π ) , ∀ t ∈ [0 , T ] , ∀ r ∈ [ r (cid:63) , R ] , and the Dirichlet boundary condition on ru ( t, r (cid:63) , θ ) = u ( t, R, θ ) = 0 , ∀ t ∈ [0 , T ] , ∀ θ ∈ [0 , π ] . We use N r and N θ number of points for the space discretization, setting∆ r = R − r (cid:63) N r and ∆ θ = 2 πN θ . We denote the full discretization by u nk,j ≈ u ( r k , θ j , t n ), r k = r (cid:63) + k ∆ r , θ j = j ∆ θ , for n = 0 , ..., N , k = 0 , · · · , N r and j = 0 , · · · , N θ . We use the second order finite difference scheme in space, to approximate the NLS Ω equation: i u n +1 k,j − u nk,j ∆ t + 12 (cid:2) D r + D θ (cid:3) u n +1 k,j + 12 (cid:2) D r + D θ (cid:3) u nk,j = − F ( u n +1 k,j , u nk,j ) , (2.5) D NLS OUTSIDE AN OBSTACLE 9 where F ( u n +1 k,j , u nk,j ) is defined in (2.2) and D r u nk,j = 1∆ r r k (cid:18) r k + u nk +1 ,j − u nk,j ∆ r − r k − u nk,j − u nk − ,j ∆ r (cid:19) , (2.6) D θ u nk,j = 1 r k u nk,j +1 − u nk,j + u nk,j − ∆ θ , (2.7) F ( u n +1 k,j , u ni,j ) = 2 p + 1 (cid:12)(cid:12)(cid:12) u n +1 k,j (cid:12)(cid:12)(cid:12) p +1 − (cid:12)(cid:12)(cid:12) u nk,j (cid:12)(cid:12)(cid:12) p +1 (cid:12)(cid:12)(cid:12) u n +1 k,j (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) u nk,j (cid:12)(cid:12)(cid:12) u n +1 k,j + u nk,j , (2.8)where r k + = r k + r k +1 , and using the convention that u n ,j = u nN r ,j = 0, u nk, = u nk,N θ and u nk, = u nk,N θ +1 from the boundary settings.To solve the above system (2.5) with (2.6) and (2.7), we consider the initial condition such that u satisfies Dirichlet boundary conditions. We typically consider a shifted Gaussian as initial condition,therefore, we define the translation parameters ( x c , y c ) such that u is smooth and vanishes to 0 nearboth the obstacle and the boundary of the computational domain.2.2. Mass and energy conservation.
The Crank-Nicholson scheme (2.1) conserves the following dis-cretized quantities: the discretized L -norm, or often referred to as the discrete mass, and the discretizedenergy, called the discrete energy, which are the discrete analogues of the mass and energy conservationsin (1.1) and (1.2).If we consider the rectangular coordinates ( x, y ), and the discretization of the Laplacian term ∆ u by thestandard five points stencil finite difference approximation (e.g., see [12]), then the inner product of (2.1)with (¯ u n +1 + ¯ u n )∆ x ∆ y , followed by considering the imaginary part, gives us the following conservationof the discrete mass: M [ u n ] = N x (cid:88) k =0 N y (cid:88) j =0 | u nk,j | ∆ x ∆ y = M [ u ] , for n ≥ , (2.9)where u nk,j := u n ( x k , y j ) , ∆ x = x k +1 − x k and ∆ y = y j +1 − y j . And N x , N y are the number of grid pointsassigned on the x,y direction respectively.Similarly, the conservation of the discrete energy in rectangular coordinates ( x, y ) is obtained by takingthe inner product of (2.1) with (¯ u n +1 − ¯ u n )∆ x ∆ y , then the real part gives E [ u n ] = 12 N x (cid:88) k =0 N y (cid:88) j =0 (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) u nk +1 ,j − u nk,j ∆ x (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) u nk,j +1 − u nk,j ∆ y (cid:12)(cid:12)(cid:12)(cid:12) − p + 1 | u nk,j | p +1 (cid:33) ∆ x ∆ y = E [ u ] , for n ≥ , (2.10)For brevity, we omit the above standard proof.In polar coordinates ( r, θ ) , the scheme (2.5) also conserves the discrete mass and energy exactly, similarlyto the above. More specifically, we define the discrete mass at t = t n by M [ u n ] = N r (cid:88) k =0 N θ (cid:88) j =0 | u nk,j | r k ∆ r ∆ θ = M [ u ] , for n ≥ . (2.11) One can see that the definition (2.11) is an analog to the mass in (1.1). In the same spirit, we define thediscrete energy as E [ u n ] = 12 N r (cid:88) k =0 N θ (cid:88) j =0 (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) u nk +1 ,j − u nk,j ∆ r (cid:12)(cid:12)(cid:12)(cid:12) r k + ∆ r ∆ θ + 12 1 r k (cid:12)(cid:12)(cid:12)(cid:12) u nk,j +1 − u nk,j ∆ θ (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ (cid:33) − p + 1 N r (cid:88) k =0 N θ (cid:88) j =0 | u nk,j | p +1 r k ∆ r ∆ θ = E [ u ] , for n ≥ . (2.12)which is the analog of the energy in (1.2).We have the following theorem: Theorem A.
The numerical scheme (2.5) conserves the discrete mass (2.11) and the discrete energy (2.12) for all n ∈ N , i.e., M [ u n ] = M [ u ] , and E [ u n ] = E [ u ] . Proof.
The proof of the mass-conservation is similar to the case in the rectangular coordinates ( x, y ) , itsuffices to take the inner product of (2.5) with (¯ u n +1 k,j + ¯ u nk,j ) r k ∆ r ∆ θ, sum up over k and j from 0 to N r ,0 to N θ respectively, and then take the imaginary part.For the energy-conservation, the proof is slightly different than the one in the rectangular coordinates( x, y ), due to the space discretization of the Laplacian in (2.6). For that, we write the scheme (2.5) for u nk,j , using (2.6), (2.7) and (2.8), to obtain i u n +1 k,j − u nk,j dt (cid:124) (cid:123)(cid:122) (cid:125) ( I ) k,j + 12 1 r k r (cid:32) r k + u n +1 k +1 ,j − u n +1 k,j ∆ r − r k − u n +1 k,j − u n +1 k − ,j ∆ r (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) ( I , ) k,j + 12 1 r k r (cid:18) r k + u nk +1 ,j − u nk,j ∆ r − r k − u nk,j − u nk − ,j ∆ r (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ( I , ) k,j + 12 1 r k u n +1 k,j +1 − u n +1 k,j + u n +1 k,j − dθ (cid:124) (cid:123)(cid:122) (cid:125) ( I , ) k,j + 12 1 r k u nk,j +1 − u nk,j + u nk,j − dθ (cid:124) (cid:123)(cid:122) (cid:125) ( I , ) k,j = − p + 1 | u n +1 k,j | p +1 − | u nk,j | p +1 | u n +1 k,j | − | u nk,j | ( u n +1 k,j + u nk,j ) (cid:124) (cid:123)(cid:122) (cid:125) ( I ) k,j (2.13)Taking the inner product of (2.13) with (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ , then the real part and summing up over k and j , yields Re N r (cid:88) k =0 N θ (cid:88) j =0 ( I ) k,j × (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ = 0 . (2.14)Re N r (cid:88) k =0 N θ (cid:88) j =0 (( I , ) k,j + ( I , ) k,j ) × (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ = N r (cid:88) k =0 N θ (cid:88) j =0 − r k + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n +1 k +1 ,j − u n +1 k − ,j ∆ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ + N r (cid:88) k =0 N θ (cid:88) j =0 r k + (cid:12)(cid:12)(cid:12)(cid:12) u nk +1 ,j − u nk,j ∆ r (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ. (2.15)Using that ¯ u nk,N θ = ¯ u nk, , u nk,N θ +1 = u nk, , u nk,N θ = u nk, and u n ,j = u nN r ,j = 0 , for all n ∈ N , we get D NLS OUTSIDE AN OBSTACLE 11 Re N r (cid:88) k =0 N θ (cid:88) j =0 ( I , ) k,j × (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ = −
12 1 r k Re N r (cid:88) k =0 N θ (cid:88) j =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n +1 k,j − u n +1 k,j − ∆ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ −
12 1 r k θ Re (cid:20) N r (cid:88) k =0 N θ (cid:88) j =0 u n +1 k,j +1 ¯ u nk,j + 2 u n +1 k,j ¯ u nk,j − u n +1 k,j − ¯ u nk,j (cid:21) ∆ r ∆ θ. (2.16)Similarly, we deduceRe N r (cid:88) k =0 N θ (cid:88) j =0 ( I , ) k,j × (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ = 12 1 r k Re Nr (cid:88) k =0 N θ (cid:88) j =0 (cid:12)(cid:12)(cid:12)(cid:12) u nk,j − u nk,j − ∆ θ (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ −
12 1 r k θ Re Nr (cid:88) k =0 N θ (cid:88) j =0 (cid:20) − ¯ u n +1 k,j +1 u nk,j − u n +1 k,j u nk,j + ¯ u n +1 k,j − u nk,j (cid:21) ∆ r ∆ θ. (2.17)By (2.17) and (2.16), we obtainRe N r (cid:88) k =0 N θ (cid:88) j =0 (( I , ) k,j + ( I , ) k,j ) × (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ = −
12 1 r k Re Nr (cid:88) k =0 N θ (cid:88) j =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n +1 k,j − u n +1 k,j − ∆ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ + 12 1 r k Re Nr (cid:88) k =0 N θ (cid:88) j =0 (cid:12)(cid:12)(cid:12)(cid:12) u nk,j − u nk,j − ∆ θ (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ. (2.18)Re Nr (cid:88) k =0 N θ (cid:88) j =0 ( I ) k,j × (¯ u n +1 k,j − ¯ u nk,j ) r k ∆ r ∆ θ = − p + 1 Nr (cid:88) k =0 N θ (cid:88) j =0 (cid:16) | u n +1 k,j | p +1 − | u nk,j | p +1 (cid:17) r k ∆ r ∆ θ (2.19)Summing up (2.14), (2.15), (2.18) and (2.19), we finally arrive at E [ u n ] = 12 N r (cid:88) k =0 N θ (cid:88) j =0 r k + (cid:12)(cid:12)(cid:12)(cid:12) u nk +1 ,j − u nk,j ∆ r (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ + 12 1 r k (cid:12)(cid:12)(cid:12)(cid:12) u nk,j +1 − u nk,j ∆ θ (cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ − p + 1 N r (cid:88) k =0 N θ (cid:88) j =0 | u nk,j | p +1 r k ∆ r ∆ θ = 12 N r (cid:88) k =0 N θ (cid:88) j =0 r k + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n +1 k +1 − u n +1 k ∆ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ + 12 1 r k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n +1 k,j +1 − u n +1 k,j ∆ θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ r ∆ θ − p + 1 N r (cid:88) k =0 N θ (cid:88) j =0 | u n +1 k,j | p +1 r k ∆ r ∆ θ = E [ u n +1 ] . (cid:3) We note that in our simulations the mass and the energy are well preserved: the relative mass-errorand energy-error are bounded by at least 10 − , at the end of the simulations at T = 20 with the timestep 10 − , as shown in Figure 2. The evolution of the relative mass and energy errors can be tracked bymax ≤ m ≤ n ( M [ u m ]) − min ≤ m ≤ n ( M [ u m ]) and max ≤ m ≤ n ( E [ u m ]) − min ≤ m ≤ n ( E [ u m ]) , (2.20) or M [ u n ] − M [ u ] M [ u ] and E [ u n ] − E [ u ] E [ u ] . (2.21) Figure 2.
Evolution of the relative mass and energy errors (2.20) in the scheme (2.5) in2 d and p = 3 . In Figure 2, we show that the error in (2.20) for the discrete mass and energy are on the order of 10 − and 10 − , correspondingly. The errors of mass and energy in the case for p = 5 are on the same order.3. The
NLS R equation In this section, we show different numerical simulations of the focusing nonlinear Schr¨odinger equationon the whole Euclidean space R . For that, we consider a bounded computational domain without theobstacle and we impose Dirichlet boundary conditions on the artificial boundary of the bounded domain,which does not affect the solution in the interior of the domain. In order to approximate the NLS R equation, we use the same time discretization, i.e, the implicit Crank-Nickolson scheme given in (2.1)with the Newton iteration to solve the nonlinear terms. In this section we give various examples that willbe considered in the next sections in order to study the influence of the obstacle on the behavior of thesolutions. Remark 3.1.
In this section, we state examples in rectangular coordinates ( x, y ) , however, in our imple-mentations for NLS Ω equation we convert them into polar coordinates ( r, θ ) . Consider the focusing cubic NLS R equation ( L -critical), with the Gaussian initial condition u ( x, y ) := u (0 , x, y ) = A e − (( x − x c ) +( y − y c ) ) e i ( ( v x · x + v y · y )) , (3.1)where v := ( v x , v y ) is the velocity vector and ( x c , y c ) is the translations. Note that, here, all parametersare the same as in Section 6.1 below, in particular, see Figure 15 for the velocity direction. Regardlessof the velocity v directions, the solution for the NLS R equation (considered on the domain without theobstacle) blows up in finite time. Snapshot at t = 0 .
75 of the solution u ( t ) to NLS R equation thatblows up in finite time is shown in Figure 3. We note that the same initial condition will have a differentevolution when an obstacle present, see Figures 24, 25.Next, we consider the focusing quintic NLS R equation, with the initial data u as in (3.1) with v = ( v x ,
0) and ( x c = − . , y c = 0) . Let us mention that, all parameters here are the same as in Section7.2 below, for comparison a reader can peak at Figure 27. In the case with no obstacle, the solution u ( t )to the NLS R equation, which is moving on the line y = 0, blows up in finite time at t = 0 .
64 and it isplotted in Figure 4 with the L ∞ -norm. On the other hand, the solution to the NLS Ω equation with theobstacle present does not blow-up in finite time, it has a completely different dynamics, furthermore, thesolitary wave does not even preserve its shape after the collision. D NLS OUTSIDE AN OBSTACLE 13
Figure 3.
Solution to the 2 d cubic NLS R with initial data u as in (3.1) at times t = 0and t = 0 .
75 moving along the line y = x on the left and middle subplots, and the timedependence of the L ∞ -norm of this solution on the right subplot. Figure 4.
Solution to the 2 d quintic NLS R with initial data u as in (3.1) at times t = 0and t = 0 .
64 moving on the line y = 0 on the left and middle subplots, and the timedependence of the L ∞ -norm of this solution on the right subplot.4. Perturbations of the soliton
We next investigate the NLS Ω evolution of a multiple of the shifted ground state, or referring to it asa perturbed soliton, of the form u ≡ u ( x, y,
0) = λ Q ( x − x c , y − y c ) , λ ∈ R , where Q is numerically constructed ground state solution to (1.3) shifted by ( x c , y c ) . The perturbation of the soliton solution to (NLS Ω ) with a ‘large mass’ initial condition, for example, λ = 1 . , leads to a blow up solution at time t = 0 .
84 with the diverging L ∞ -norm. Recall that we usethe Newton iteration to solve the implicit scheme (2.1) and to reach the desired accuracy. Thus, it isdifficult to approach the blow-up time while maintaining the convergence of the Newton iteration (2.3).For that, we need to run the scheme with a more refined mesh in order to maintain the convergence of(2.3). This is challenging to handle in the 2 d non-radial case. For practical purposes we consider that a solution blows up in finite time when its height ( L ∞ -norm) is, for example, 3 times higher than the initialone. Figure 5.
Solution to (NLS Ω ) with u ( x, y,
0) = 1 . Q ( x + 4 . , y ) at t = 0 .
84 on the leftand its L ∞ norm depending on time on the right.We now look at the initial condition of the perturbed soliton with the mass smaller than that of theground state, i.e., u (0 , x, y ) = 0 . Q ( x + 4 . , y ). A snapshot of the corresponding solution of the NLS Ω equation at time t = 1 . L ∞ norm is shown in Figure 6. In the presentsituation, we see that the L ∞ norm is monotonically decreasing with a definite negative slope. Therefore,we conclude that this solution disperses in a long run, as expected for perturbations with smaller massthan that of the soliton. Nevertheless, to confirm our expectations we run this example with the sameinitial condition for longer times and the next Figure 7 shows that the L ∞ -norm keeps decreasing toward0 . Figure 6.
Solution to (NLS Ω ) with u ( x, y,
0) = 0 . Q ( x + 4 . , y ) at t = 1 . L ∞ norm for 0 < t < . u similar to (3.1), sinceit has a faster decay than the ground state (though both have exponential decays), which ensures that D NLS OUTSIDE AN OBSTACLE 15
Figure 7.
The L ∞ -norm for the solution in Figure 6 for 0 < t < . the simulations close to the obstacle satisfy Dirichlet boundary condition, and thus, even a slightly fasterdecay makes computations easier. Moreover, in order to study various interactions with the obstacle, weconsider initial data u with minimal possible distance d (cid:63) to the obstacle so that u still satisfies Dirichletboundary. In what follows, we study the dependence of the interaction on the distance between the initialdata and the obstacle. 5. Dependence on the distance
From now on we study both the 2 d cubic and quintic NLS Ω equations ( p = 3 , u such that the distance d between the obstacle and the initialcondition is larger than the minimal distance d ∗ . We take a shifted Gaussian initial condition similar to(3.1), u (0 , x, y ) = A e − (( x − x ) +( y − y ) ) , (5.1)where ( x , y ) - the translation parameters such that d >> d ∗ , thus, u (0 , x, y ) is smooth and satisfiesDirichlet boundary conditions. We denote by u the initial data with the phase e i ( ( v x · x + v y · y )) , i.e., u ( x, y ) = A e − (( x − x ) +( y − y ) ) e i ( ( v x · x + v y · y )) , (5.2)where v := ( v x , v y ) is the velocity vector, which governs the movement of this initial condition. Figure 8shows different directions of solitary wave propagation with the velocity (cid:126)v. Figure 8.
If the solition is far from the obstacle ( d >> d ∗ ), then the blow-up occurs inany direction of the initial velocity (cid:126)v shown on the picture. We start here with the cubic NLS Ω equation and take the initial condition (5.2) with large enoughmass and d >> d ∗ . Then the corresponding solution to (2.1) blows up in finite time before reaching orinteracting with the obstacle for any direction of the velocity vector v, see Figure 9. We will also studythe case when d ≡ d ∗ and the solution will concentrate in its (blow-up) core after the obstacle, for thesame initial data but for a different velocity direction. We will also study the influence of the obstaclewhen there is an interaction between the traveling wave and the obstacle. In Section 6.1, we consider theweak interaction for the cubic (NLS Ω ) equation ( L -critical case) and in Section 7.1 we study the stronginteraction. We will observe that in those cases the solution may have a different behavior on a longertime interval. Figure 9.
Profile of the initial data u and a snapshot of the solution u ( t ) to (2.1) (thewhite region represents the obstacle) at times t = 0 and 0 .
51 with d >> d ∗ on the left andmiddle, the L ∞ -norm depending on time on the right.Next, we consider the 2 d quintic NLS Ω equation and take the initial condition (5.2) with large mass and d >> d ∗ . In the following scenario, we fixed all parameters ( A and v = ( v x , x , y ) . We vary the vertical translation parameters y , as shown in the following Figure 10. Figure 10.
The direction of solitary waves moving in lines y = 5 or y = 2 with d >> d ∗ . D NLS OUTSIDE AN OBSTACLE 17
A snapshot of the corresponding solution to (2.1) is plotted in Figure 11. As in the previous example,the solution blows up in finite time before the obstacle for x large. Later we also consider d ≡ d ∗ with the same setting of the initial data u and parameters, i.e., we fix the variables A = 1 .
25 andvelocity v = (15 ,
0) and study the interaction in Section 6.2 and 7.2. That will lead to the weak or stronginteraction for the quintic NLS Ω equation ( L -supercritical case). Figure 11.
Solution to (2.1) with the initial condition u and d >> d ∗ ; the initial datais on the left subplot, the solution that blows up in finite time at t = 0 .
63 before theobstacle is on the middle subplot (the white region represents the obstacle). Right: thetime dependence of the L ∞ -norm.6. Weak interaction with an obstacle
The L -critical case. We return to the cubic NLS Ω setting ( p = 3) , which is L -critical. We studythe time evolution of the initial condition u = A e − (( x − x c ) +( y − y c ) ) e i ( ( v x · x + v y · y )) , (6.1)where ( x c , y c ) is the translation, v = ( v x , v y ) is the velocity vector and A is the amplitude. In thefollowing simulation, we fix A = 2 .
25, and also ( x c , y c ), but we vary the direction of the velocity vector.We choose x c = − . y c = − . u is smooth and satisfies Dirichlet boundary conditions.In the following simulation, we consider the two scenarios in Figure 12. Figure 12.
The direction of the movement of the solution in Example 6.1.
We start with the initial datum u described above with (cid:126)v = ( v x , , v x = 15 . We observe that thesolution blows up at time t = 0 .
52. It does not interact with the obstacle; its behavior is the same as itwould be of a solitary wave on the whole space.
Figure 13.
The initial condition u from (6.1) (left); the time evolution at t = 0 . L ∞ -norm (right).The same initial condition (6.1) but with the velocity vector v that is perpendicular to the directionin the previous example is shown in Figure 12. The velocity vector is v = ( v x , v y ) = (0 , . We observethat the solution blows up at the same time.
Figure 14.
The initial data u (6.1) (left); its time evolution to (2.1) at t = 0 .
52 (middle);the time dependence of the L ∞ -norm (right).In our third example, we take the initial datum u (6.1) with the velocity (cid:126)v that has a different directionbut has the same magnitude | (cid:126)v | , as in the previous two examples: we choose v = ( v x , v y ) and v = ( v y , v x )as shown on Figure 15. D NLS OUTSIDE AN OBSTACLE 19
Figure 15.
The direction of movement of the solution in the next examples.We choose v = (15 ,
9) such that the solution has a small interaction with the obstacle. After thecollision, we observe that the solution has almost the same behavior (as the previous example with aweak interaction see Figure 22), i.e., it blows up with slightly dispersive reflection part, preserving theshape of the soliton, similar to the two previous cases. The solution blows up in finite time t = 0 . L ∞ -norm hasagain a slight perturbation (or a small oscillation), however, afterwards it continues to increase and isperturbed less in the overall growing of the L ∞ -norm, compared to the perturbation in the previous caseshown in Figure 22. We also provide snapshots of the behavior of the solution for different time steps for v = (9 , , see Figure 17. Figure 16.
Solution to (2.1) with initial condition u (6.1) and velocity v = (15 ,
9) attime t = 0 . , moving on the line y = x (left); the time dependence of L ∞ -norm (right). Figure 17.
Snapshots of the behavior of the solution u to (2.1) with v = (9 ,
15) movingon the line y = x .In our 4 th example here, we take the same initial condition (6.1) for u but with the velocity vector v = (12 ,
15) and v = (15 , . For v = (12 , , a snapshot of the solution at time t = 1 . L ∞ -norm depending on time, which appears to growquite fast at the beginning of the simulation but after the collision it starts to decrease monotonically.This solution disperses, or in other words, it becomes a scattering solution. Thus, the obstacle arreststhe blow-up. This is a different behavior compared to the previous examples, where the solutions weretransmitted almost with the same shape after the interaction and the soliton core was preserved. Unlikethe previous examples, the collision of the solution with the obstacle here creates reflected waves, whichthen disperse the solution. The reflection causes the loss of the mass in the main part of the solution,which arrests the blow-up in finite time unlike the examples above, where the reflection does not affectthe blow-up of the solution and only influences (delays) the blow-up time. In this case the interactionbetween the soliton and the obstacle has a substantial influence on the behavior of the solution, which is acompletely new dynamics compared to the dynamics on the whole space. For better understanding of thedynamics, we provide snapshots of the behavior of the solution for different time steps for v = (15 , , see Figure 19. D NLS OUTSIDE AN OBSTACLE 21
Figure 18.
Solution to (2.1) with initial condition u and velocity v = (15 ,
12) at time t = 1 . y = x (left), the time dependence of the L ∞ -norm of thesolution (right). Figure 19.
Snapshots of the behavior of the solution u to (2.1) with v = (15 ,
12) movingon the line y = x . In the following simulation, we study the behavior of various examples of solutions to the cubic NLS Ω equation, depending on the interaction between the solution and the obstacle. We consider the sameinitial condition as above, u = A e − (( x − x c ) +( y − y c ) ) e i ( ( v x · x + v y · y )) , but we vary the direction of the velocity vector ( v x , v y ) and we fix the following parameters as follows: A = 2 . , x c = − . , y c = − . . We record the results of our simulations in Table 1.( v x , v y ) Discrete mass Discrete energy Behavior of the solution Type of interaction(15 ,
0) 15.9043 442.9353 Blow-up at t ≈ .
52 no-interaction(0 ,
15) 15.9043 442.9353 Blow-up at t ≈ .
52 no-interaction(15 ,
8) 15.9043 570.4814 Blow-up at t ≈ .
56 weak-interaction(8 ,
15) 15.9043 570.4814 Blow-up at t ≈ .
56 weak-interaction(9 ,
15) 15.9043 604.0182 Blow-up up t ≈ .
57 weak-interaction(15 ,
9) 15.9043 604.0182 Blow-up at t ≈ .
57 weak-interaction(10 ,
15) 15.9043 641.4737 Blow-up at t ≈ .
63 weak-interaction(15 ,
10) 15.9043 641.4737 Blow-up at t ≈ .
63 weak-interaction(15 ,
12) 15.9043 728.1404 Scattering weak-interaction(12 ,
15) 15.9043 728.1404 Scattering weak-interaction(15 ,
15) 15.9043 887.5015 Scattering strong-interaction
Table 1.
Different velocity directions (cid:126)v = ( v x , v y ) and the corresponding behavior of thesolution u ( t ) with the discrete mass and energy (the value of energy differs due to phase).6.2. The L -supercritical case. We consider the 2 d quintic NLS Ω equation ( p = 5) , which is L -supercritical. Again, we try to carefully examine the interaction between the obstacle and the solution.For that, we take a shifted Gaussian u = A e − (( x − x c ) +( y − y c ) ) as the initial condition with a givenvelocity and amplitude, i.e., u ≡ u (0 , x, y ) := A e − (( x − x c ) +( y − y c ) ) e i ( ( v x .x + v y .y )) . (6.2)In the following simulations, we fix A and the velocity v , but we change the translation parameters. Recallthat, we choose x c and y c such that u is smooth and satisfies Dirichlet boundary conditions, see Figure 10.We start with an example, where there is no interaction in order to compare the behavior of the solutionfor different scenarios later, especially when there will be a strong interaction. For that, we consider theinitial data u from (6.2) with A = 1 . , x c = − . , y c = 5 , and v = (15 , , (6.3)which can be seen on the left of Figure 20. The middle subplot shows that the corresponding solution of(2.1) blows up in finite time at t = 0 .
65 with the diverging L ∞ -norm. Snapshots of the solution in timeare plotted in Figure 21. We observe that the solution blows up in finite time and there is no interactionbetween the solution and the obstacle. D NLS OUTSIDE AN OBSTACLE 23
Figure 20.
Solution to (2.1) with u from (6.3) (left) close to blow-up time (middle),time dependence of the L ∞ -norm (right). Figure 21.
Snapshots of the evolution of u from (6.3) in time t = 0 , t = 0 .
38 and t = 0 . . Next, we take the same initial data u as in the previous example (6.3) with y c = 2 as shown in Figure10. In this case, we expect that the traveling wave solution has some weak interaction with the obstacle. Figure 22.
Solution to (2.1) with u from (6.3) with y c = 2. Left: snapshot at time t = 0 . . Right: the time dependence of the L ∞ -norm. We observe that with this weak interaction the solution still blows up in finite time at t = 0 .
66 butthe blow-up time is delayed compared to the previous case, where there was no interaction between thesolution and the obstacle, see Figure 22. Moreover, we observe a slight perturbation of the growth in the L ∞ -norm: at the collision, the amplitude of the solution starts decreasing but after the weak interaction,the solution is back to the concentration leading to the blow-up. This can be explained by the appearanceof small reflected waves after the collision, which scatter at the end of the simulation. They can be seen inthe snapshots of the solution in Figure 23 with the view onto the xy -plane and zooming near the obstacle. Figure 23.
Snapshots of the time evolution of the solution u with initial u as in (6.3)with y c = 2, which eventually blows up in finite time.Next, we summarize the behavior of the solution to the quintic NLS Ω equation, depending on the initialparameters. We give various values to the space translation ( x c , y c ) in the initial condition u ≡ u (0 , x, y ) := A e − (( x − x c ) +( y − y c ) ) e i ( ( v x .x + v y .y )) , and we fix the following parameters: A = 1 . , v x = 15 , v y = 0 . The results are given in Table 2.
D NLS OUTSIDE AN OBSTACLE 25 ( x c , y c ) Discrete mass Discrete energy Behavior of the solution Type of interaction( − . ,
5) 4.9087 138.9766 Blow-up at t ≈ .
65 no interaction( − . ,
4) 4.9087 139.2859 Blow-up at t ≈ .
68 no interaction( − . ,
3) 4.9087 139.4553 Blow-up at t ≈ .
65 weak-interaction( − . ,
2) 4.9087 139.5022 Blow-up at t ≈ .
66 weak-interaction( − . , .
5) 4.9087 139.4946 Blow-up at t ≈ .
51 weak-interaction( − . ,
1) 4.9087 139.4784 Blow-up at t ≈ .
41 weak-interaction( − . , .
5) 4.9087 139.4636 Scattering weak-interaction( − . ,
0) 4.9087 139.4578 Scattering strong-interaction( − . , − .
5) 4.9087 139.4636 Scattering weak-interaction( − . , −
1) 4.9087 139.4784 Blow-up at t ≈ . − . , − .
5) 4.9087 139.4946 Blow- up at t ≈ . − . , −
2) 4.9087 139.0924 Blow-up at t ≈ .
63 weak-interaction
Table 2.
Different translation parameters ( x c , y c ) and the corresponding behavior of thesolution u ( t ) with the discrete mass and energy. Note that a tiny difference in the valuesof the discrete energy for different ( x c , y c ) results from the different density of the meshgrid, this is due to the fact that the polar coordinates ( r, θ ) form a uniform mesh, however,( x, y ) = ( r cos θ, r sin θ ) will not be uniform. Nevertheless, the discrete energy is conservedfrom the start, i.e., E [ u n ] = E [ u ], in each simulation.7. Strong interaction with an obstacle
The L -critical case. We now consider a direct interaction of the solution with an obstacle, whichwe term as a strong interaction, starting with the L -critical case. The depiction of the velocity directionand location is in Figure 24. We fix the cubic NLS Ω equation with the same initial condition (6.2) and Figure 24.
The direction of movement of the solution u on the line y = x and the samedirection of the outward normal vector.we take the velocity v = (15 ,
15) ( A = 2 . , x c = − . y c = − . u is smooth and satisfies Dirichlet boundaries condition. Note that the solution u is moving inthe line y = x, i.e, in the same direction of the normal vector as described in Figure 24. Figure 25.
The initial data u at t = 0 (left); the corresponding solution u to (2.1) attime t = 1 . L ∞ norm (left). Figure 26.
Snapshots of the behavior of the solution u to (2.1) from Figure 25. D NLS OUTSIDE AN OBSTACLE 27
We observe that the strong interaction has a substantial influence on the dynamics of the solution.Recall that, in the weak interaction case for a similar example in § L -critical) the solution blows upin finite time. Here we observe a scattering behavior here, the solution splits into several bumps with adispersive backward reflected waves. The obstacle transforms the blow-up behavior into scattering.7.2. The L -supercritical case. In the 2 d quintic NLS Ω equation ( p = 5), we also investigate the stronginteraction between the obstacle and the solution. Recall that we call an interaction strong , if the solutionis moving in the same direction as the outward normal vector of the obstacle, see Figure 27. Figure 27.
The direction of movement of the solution, on the line y = 0 with outwardnormal vector.We consider the same initial data, i.e., the shifted Gaussian (6.2), with the same phase as for the quinticNLS Ω equation described in subsection 6.2 with y c = 0 ( A = 1 . , x c = − . v = ( v x ,
0) are fixedparameters). In the present situation, the solution is moving on the line y = 0 , i.e., in the same directionof the outward normal vector to the obstacle. The solitary wave hits the obstacle straight on, causing astrong interaction between the wave and the obstacle, see Figure 28. Figure 28.
Solution to (2.1) with the initial data u (6.2): the first subplot is the initialdata at t = 0 ; middle subplot is the solution u ( t ) in time, moving on the line y = 0; thelast subplot is the time dependence of the L ∞ -norm of the solution. In this case, the solution scatters and does not preserve the shape of the original solitary wave. After thecollision, the solitary wave solution splits into several waves and behaves as a sum of two or more solitonswith a backward reflection, see Figure 29. We observe also that the leading reflected wave has a dispersivebehavior. Moreover, one can see that the presence of the obstacle completely prevents blow-up. Beforethe interaction, the L ∞ -norm of the solution starts increasing, indicating a possible blow-up behavior,however, after the collision time the amplitude of the solution decreases toward 0, which confirms thedispersion of the solution in a long term, thus, scattering. Figure 29.
Snapshots of the behavior of the solution u to (2.1) with ( x, y ) − view.8. Conclusion
In this work we initiated a numerical study of how the behavior of solutions can change in a presence ofa convex smooth obstacle. We observe that the interaction between a solitary wave and the obstacle hasan influence on the overall behavior of the solution to the NLS Ω equation, which depends on the directionof the velocity vector (cid:126)v , the initial distance to the obstacle and the translation parameters ( x c , y c ). Thepresence of the obstacle yields strong, weak or no interaction. We observed in Sections 6.2 and 6.1 that D NLS OUTSIDE AN OBSTACLE 29 even a small interaction between the obstacle and the solution has some influence on the dynamics ofthe equation (at the least, on the blow-up time). Moreover, we conclude that the strong interaction hasa significant effect on the behavior of the solution, for example, instead of approaching a solitary wavesolution with a single bump, the shape of the solution drastically changes, splitting it into several bumps,which eventually scatter. In such a case, the appearance of the reflection waves due to the presence ofthe obstacle with Dirichlet boundary conditions prevents the solution from blowing up in finite time.Furthermore, this backward reflection has always a dispersive character, this might be the reason whythe solution behaves as a multi-soliton solution in a long run after a strong interaction. For a weakinteraction, i.e., when the solution preserves the shape as a traveling solitary wave, the solution behavesas either a solitary wave solution, constructed in [31] (which exists for all positive times), or as the oneshown in [30] (see also [29]), a finite time blow-up solution.
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Email address : [email protected] LAGA, UMR 7539, Institut Galil´ee, Universit´e Sorbonne Paris Nord.
Email address : [email protected] Department of Mathematics & Statistics, Florida International University, Miami, FL 33199, USA.
Email address : [email protected]@fiu.edu